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Find an orthogonal basis for R^3containing the vector 12-2
Related Topics
Wize University Linear Algebra Textbook > Orthogonality
Orthonormal Basis and Gram-Schmidt Process
5 Activities
Find an orthogonal basis for
R
3
R^3
R
3
containing the vector
[
1
2
−
2
]
\begin{bmatrix}1\\2\\-2\end{bmatrix}
1
2
−
2
{
[
1
2
−
2
]
,
[
0
1
1
]
,
[
0
1
−
1
]
}
\left\{\begin{bmatrix}1\\2\\-2\end{bmatrix},\begin{bmatrix}0\\1\\1\end{bmatrix},\begin{bmatrix}0\\1\\-1\end{bmatrix}\right\}
⎩
⎨
⎧
1
2
−
2
,
0
1
1
,
0
1
−
1
⎭
⎬
⎫
{
[
1
2
−
2
]
,
[
0
1
1
]
,
[
1
−
1
1
]
}
\left\{\begin{bmatrix}1\\2\\-2\end{bmatrix},\begin{bmatrix}0\\1\\1\end{bmatrix},\begin{bmatrix}1\\-1\\1\end{bmatrix}\right\}
⎩
⎨
⎧
1
2
−
2
,
0
1
1
,
1
−
1
1
⎭
⎬
⎫
{
[
1
2
−
2
]
,
[
0
1
1
]
,
[
4
−
1
1
]
}
\left\{\begin{bmatrix}1\\2\\-2\end{bmatrix},\begin{bmatrix}0\\1\\1\end{bmatrix},\begin{bmatrix}4\\-1\\1\end{bmatrix}\right\}
⎩
⎨
⎧
1
2
−
2
,
0
1
1
,
4
−
1
1
⎭
⎬
⎫
{
[
1
2
−
2
]
,
[
0
1
1
]
,
[
2
1
−
1
]
}
\left\{\begin{bmatrix}1\\2\\-2\end{bmatrix},\begin{bmatrix}0\\1\\1\end{bmatrix},\begin{bmatrix}2\\1\\-1\end{bmatrix}\right\}
⎩
⎨
⎧
1
2
−
2
,
0
1
1
,
2
1
−
1
⎭
⎬
⎫
None of the above
I don't know
Check Submission
More Orthonormal Basis and Gram-Schmidt Process Questions:
Span
Let
B
:
=
{
(
1
,
1
,
0
,
2
)
,
(
0
,
1
,
0
,
−
1
)
,
(
0
,
0
,
1
,
1
)
}
B:=\{(1,1,0,2),(0,1,0,-1),(0,0,1,1)\}
B
:=
{(
1
,
1
,
0
,
2
)
,
(
0
,
1
,
0
,
−
1
)
,
(
0
,
0
,
1
,
1
)}
, and
U
:
=
s
p
a
n
B
U:=spanB
U
:=
s
p
an
B
.
a)
Find a pairwise orthogonal set of nonzero vectors
O
O
O
so that
U
=
s
p
a
n
O
U=spanO
U
=
s
p
an
O
.
b)
Write the Fourier expansion of
(
0
,
0
,
1
,
0
)
(0,0,1,0)
(
0
,
0
,
1
,
0
)
in terms of the basis
O
O
O
.
19.4F_Final_Builder_Ch_17.2_Orthogonal_Complement_$\tkco{eg1}$_$\key{Final}$_Builder_$\tkcth{17.2.}\tkcf{1}$_$\key{Quiz}$
Show that the set of vectors
{
[
1
2
1
]
[
−
1
1
−
1
]
[
−
3
0
3
]
}
\left\{ \colvecth{1}{2}{1} \colvecth{-1}{1}{-1} \colvecth{-3}{0}{3} \right\}
⎩
⎨
⎧
1
2
1
−
1
1
−
1
−
3
0
3
⎭
⎬
⎫
is a basis for
R
3
\mathbb{R}^3
R
3
using *only* the fact that the vectors are all perpendicular to one another
(
i.e.
no points would be awarded for using Gaussian elimination).
Consider the subspace
W
=
{
(
x
,
y
,
z
)
∈
R
3
∣
x
−
y
+
3
z
=
0
}
W=\Big\{(x,y,z)\in \mathbb{R}^3 \; | \; x-y+3z=0 \Big\}
W
=
{
(
x
,
y
,
z
)
∈
R
3
∣
x
−
y
+
3
z
=
0
}
of
R
\mathbb{R}
R
.
Find a basis for
W
W
W
that contains the vector
v
⃗
=
[
0
3
1
]
\vec{v}=[0 \; 3 \; 1]
v
=
[
0
3
1
]
.
Find orthogonal complement of
W
W
W
in
R
3
\mathbb{R}^3
R
3
.
Gram-Schmidt process
Consider the basis
B
=
{
(
1
,
3
)
,
(
2
,
1
)
}
B=\{(1,3),(2,1)\}
B
=
{(
1
,
3
)
,
(
2
,
1
)}
of
R
2
\mathbb{R}^2
R
2
Use the Gram-Schmidt process to change
B
B
B
into an orthonormal basis
B
′
B'
B
′
Gram-Schmidt process
Use the Gram-Schmidt proces to find an orthonormal basis for
U
=
Span
{
(
1
,
2
,
1
,
2
)
,
(
3
,
3
,
3
,
4
)
,
(
7
,
6
,
5
,
3
)
}
U=\text{Span}\{(1,2,1,2),(3,3,3,4),(7,6,5,3)\}
U
=
Span
{(
1
,
2
,
1
,
2
)
,
(
3
,
3
,
3
,
4
)
,
(
7
,
6
,
5
,
3
)}
Practice: Gram-Schmidt Process
Use the Gram-Schmidt process to create an orthonormal basis for the subspace
W
W
W
of
R
4
\mathbb{R}^4
R
4
:
W
=
Span
(
{
u
⃗
1
,
u
⃗
2
,
u
⃗
3
}
)
W=\text{Span}\Big(\{\vec u_1,\vec u_2,\vec u_3\}\Big)
W
=
Span
(
{
u
1
,
u
2
,
u
3
}
)
where
u
⃗
1
,
u
⃗
2
,
u
⃗
3
\vec u_1,\vec u_2, \vec u_3
u
1
,
u
2
,
u
3
are the columns of the matrix:
A
=
[
1
3
6
1
1
2
1
3
4
1
1
0
]
A=\left[\begin{array}{rrr} 1&3&6\\ 1&1&2\\ 1&3&4\\ 1&1&0 \end{array}\right]
A
=
1
1
1
1
3
1
3
1
6
2
4
0
Practice Question: Gram-Schmidt
Find an orthonormal basis of
n
u
l
l
(
A
)
null(A)
n
u
l
l
(
A
)
, where
A
=
[
−
1
2
3
1
2
−
4
5
−
1
]
A=\left[\begin{array}{rrrr}-1&2&3&1\\2&-4&5&-1\end{array}\right]
A
=
[
−
1
2
2
−
4
3
5
1
−
1
]
.
Orthonormal Basis and Gram-Schmidt Process
Example: Gram-Schmidt Process
Let
U
=
span
{
(
2
,
1
,
−
1
)
,
(
−
5
,
1
,
3
)
,
(
1
,
1
,
9
)
}
U=\text{span}\{(2,1,-1),(-5,1,3),(1,1,9)\}
U
=
span
{(
2
,
1
,
−
1
)
,
(
−
5
,
1
,
3
)
,
(
1
,
1
,
9
)}
be a subspace of
R
3
\mathbb{R}^3
R
3
.
Use the Gram-Schmidt process to find an orthonormal basis for
U
U
U
.
Orthonormal Basis
Example: Orthonormal Sets
Part A)
Show that
{
(
1
,
0
,
1
,
0
)
,
(
0
,
1
,
0
,
1
)
,
(
−
1
,
0
,
1
,
0
)
,
(
0
,
−
1
,
0
,
1
)
}
\{(1,0,1,0),(0,1,0,1),(-1,0,1,0),(0,-1,0,1)\}
{(
1
,
0
,
1
,
0
)
,
(
0
,
1
,
0
,
1
)
,
(
−
1
,
0
,
1
,
0
)
,
(
0
,
−
1
,
0
,
1
)}
is an orthogonal set.
Basis of Orthogonal Complement
Find an orthonormal basis for the orthogonal complement of the intersection of the planes
2
x
+
y
−
z
=
0
2x+y-z=0
2
x
+
y
−
z
=
0
and
4
x
−
3
y
=
0
4x-3y=0
4
x
−
3
y
=
0
in
R
3
\mathbb{R}^3
R
3
.
If we apply the Gram-Schmidt process to the vectors
(
3
,
0
,
4
,
0
)
(3, 0, 4, 0)
(
3
,
0
,
4
,
0
)
,
(
−
1
,
0
,
7
,
0
)
(−1, 0, 7, 0)
(
−
1
,
0
,
7
,
0
)
, and
(
0
,
1
,
3
,
0
)
(0, 1, 3, 0)
(
0
,
1
,
3
,
0
)
in that order, to obtain the orthonormal vectors
u
1
,
u
2
,
u
3
u_1, u_2, u_3
u
1
,
u
2
,
u
3
in that order, then what is
u
3
u_3
u
3
?
Projection onto a Subspace
Calculate
p
r
o
j
s
p
a
n
{
(
1
,
1
,
0
,
1
)
,
(
1
,
0
,
1
,
1
)
}
(
4
,
−
2
,
−
1
,
−
1
)
proj_{span\{(1,1,0,1),(1,0,1,1)\}}(4,-2,-1,-1)
p
r
o
j
s
p
an
{(
1
,
1
,
0
,
1
)
,
(
1
,
0
,
1
,
1
)}
(
4
,
−
2
,
−
1
,
−
1
)
.
B
=
{
[
1
1
−
1
]
,
[
1
0
1
]
,
[
1
−
2
−
1
]
}
B=\left\{\begin{bmatrix}1\\1\\-1\end{bmatrix},\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}1\\-2\\-1\end{bmatrix}\right\}
B
=
⎩
⎨
⎧
1
1
−
1
,
1
0
1
,
1
−
2
−
1
⎭
⎬
⎫
is an orthogonal basis of
R
3
R^3
R
3
. Which of the following is the coordinate vector of
[
3
1
1
]
\begin{bmatrix}3\\1\\1\end{bmatrix}
3
1
1
relative to the basis?